杨辉三角形中的几个加法关系

The Hockey Stick Theorems in Pascal and Trinomial Triangles

We have found some patterns in some triangles.

Hyperbolic Pascal triangles

In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearence of some binary recurrences in a fixed hyperbolic triangle are investigated.

A Novel Fibonacci Pattern in Pascal's Triangle

The Fibonacci sequence is obtained as weighted sum along the rows in the Pascal triangle by choosing a periodic up-and-down pattern of weights from the set $\{-1,-\frac{1}{2},0, \frac{1}{2}, 1\}$. A graphical illustration of this identity shows a novel "`beautiful"' Fibonacci pattern.

关于帕斯卡三角形的注记

Remark on Pascal's Triangle

本文利用库默尔定理，通过初等练习解释了帕斯卡三角形模$p$时的分形结构，核心在于进位条件与整除性的等价关系。

Honeycombs in the Pascal triangle and beyond

In this paper we present a geometric approach to discovering some known and some new identities using triangular arrays. Our main aim is to demonstrate how to use the geometric patterns (by Carlitz), in the Pascal and Hosoya triangles to rediscover some classical identities and integer sequences. Therefore, we use new techniques in classical settings which then provide a new perspective in undergraduate research.

Binomial Coefficients in a Row of Pascal's Triangle from Extension of Power of Eleven: Newton's Unfinished Work

The aim of this paper is to find a general formula to generate any row of Pascal's triangle as an extension of the concept of $\left(11\right)^{n}$. In this study, the visualization of each row of Pascal's triangle has been presented by extending the concept of the power of 11 to the power of 101, 1001, 10001, and so on. We briefly discuss how our proposed concept works for any $n$ by inserting an appropriate number of zeros between $1$ and $1$ (eleven), that is the concept of $\left(11\right)^{n}$ has been extended to $\left(1\Theta1\right)^{n}$, where $Θ$ represents the number of zeros. We h…